House Republicans appeared Wednesday to be rallying around an $85 billion deal to avert ­another government shutdown, brushing off complaints from their right wing that the effort would trade immediate spending cuts for less-certain long-term savings.

After conducting a series of closed-door briefings and polling members during a late-afternoon vote, GOP leadership aides confidently predicted that the deal struck between House Budget Committee Chairman Paul Ryan (R-Wis.) and Senate Budget Committee Chairman Patty Murray (D-Wash.) would sail through the House when it comes to a vote Thursday.

The moment seemed to mark a potentially significant shift by House Republicans away from the uncompromising confrontation of recent years and toward a determined era of more functional governance. After multiple standoffs and threatened defaults and one actual shutdown, polls show that the Republican brand has been badly damaged among voters, and even some of the most conservative Republicans said they were ready for a breather.

“That’s the only reason” to vote for the budget deal — “to calm things down,” said Rep. Cynthia M. Lummis (R-Wyo.), who said she was undecided about the measure. “To reset the debate is a great way to describe it.”

The only sign of trouble came from arch-conservatives who routinely buck House Speaker John A. Boehner (R-Ohio). But they acknowledged that their government-shrinking priorities had been abandoned in favor of Republican concerns about the Pentagon budget and political fallout from another shutdown.

“This bill is not designed to get our vote,” said Rep. Mick Mulvaney (R-S.C.), an outspoken renegade first elected in the GOP landslide of 2010. “It’s designed to pass with a bipartisan vote of the House.”

Independent conservative groups with a proven ability to sway Republicans continued to batter the agreement, which would cancel half of the sharp agency cuts known as the sequester in the current fiscal year and replace them with a mix of fees and other savings. On Wednesday, Chris Chocola, president of the conservative Club for Growth, complained that the plan “swaps debt reduction today and next year for the dubious promise of debt reduction a decade from now.”

But Boehner fired back during a brief news conference Wednesday, suggesting that the Club for Growth, Heritage Action for America and other groups were vilifying a reasonable proposal to fuel their own fundraising campaigns.

“They’re using our members and they’re using the American people for their own goals,” Boehner said. “This is ridiculous.”

Glen Bolger, a leading Republican pollster, said the budget deal puts House Republicans at a crossroads: They can pass a compromise that represents progress toward the goal of reducing the size of government, or they can reject the deal as insufficiently conservative and cement the party’s reputation for obstruction.

“We are either a party that is serious about governing when control of Washington is split, or we are an unserious party that doesn’t care about realistic incremental gains, only about unrealistically getting the whole ball of wax, which will never happen as long as there is a Democratic president and a Democratic [Senate] majority leader,” Bolger said. “We have to stop being the dysfunctional equivalent of the Washington Redskins.”

Though they expect a strong Republican vote, GOP leadership aides said they were likely to need a sizable contingent of Democrats to push the measure across the finish line. Given President Obama’s strong support for the budget deal, Democrats, too, were expected to vote for the measure in large numbers.

That margin may be trimmed somewhat, key Democrats said, by a last-minute decision by Republicans to add an amendment to benefit doctors who see Medicare patients. Democrats support what is known as the “doc fix,” which would postpone for three months a 24 percent cut in Medicare re­imbursement rates set to hit providers in January. But they protested a decision Wednesday by GOP leaders to help doctors while refusing to extend unemployment insurance for the long-term unemployed, which is set to expire at the end of the month, cutting off 1.3 million people.

“It’s unconscionable,” said Rep. Sander M. Levin (Mich.), the senior Democrat on the House Ways and Means Committee. “Are we going to leave here saying we’ll reimburse physicians and cut off people who have been left out of a job through no fault of their own?”

Rep. Chris Van Hollen (Md.), the lead budget negotiator for House Democrats, said the wrinkle adds “a new dynamic that could upset the apple cart” for the painstakingly crafted budget deal.

Other Democrats said the deal was in no real danger. And aides to Senate Majority Leader Harry M. Reid (Nev.) said Reid himself helped negotiate the addition of the doc fix amendment to speed year-end passage of that measure.

Meanwhile, in a speech on the Senate floor, Reid dismissed the effort by House Democrats to extend unemployment benefits before the Christmas break, saying he would “push here after the first of the year for an extension of emergency unemployment insurance.”

Reid said he was disappointed that jobless benefits were not included in the budget deal. But he added: “Neither side got everything it wanted in these negotiations.”

In an official estimate released Wednesday, the nonpartisan Congressional Budget Office said the budget measure would reduce deficits by $85 billion over the next decade by reducing taxpayer contributions to federal-worker pensions, requiring private companies to pay higher premiums for federal insurance of their pensions and increasing security fees for airline travelers, among other changes.

The proposal also calls for small but symbolic savings from other policies, such as helping the Treasury Department prevent prison inmates from receiving improper tax payments.

Of that $85 billion, the CBO said $62 billion would go toward rolling back the sequester in fiscal 2014 and 2015, with the Pentagon and domestic programs benefiting equally. The rest of the savings, roughly $23 billion, would go toward trimming deficits projected to exceed $6 trillion over the next decade.

If the House approves the package as expected, the Senate plans to take it up next week before leaving town for its Christmas recess. Members of the House and Senate appropriations committees would then work over the holidays to prepare funding bills for individual agencies, which are likely to be combined into a single measure known as an omnibus.

While the agreement between Ryan and Murray lessens the odds of another shutdown when a temporary spending bill expires Jan. 15, the omnibus must pass before that deadline to keep the government open.

That would give Congress relatively clear sailing on the budget until the fall of 2015 — except for the need to raise the federal debt limit, probably sometime in late February or early March.

Asked Wednesday what Republicans should demand in exchange for lifting the debt limit, a gathering of conservative lawmakers organized by the Heritage Foundation threw out a range of ideas, including an agreement to advance an overhaul of the tax code.

But Rep. Raúl R. Labrador (R-Idaho) — demoralized by the latest budget deal and by GOP leaders’ public vows never to default on the national debt — said he doubted Republicans would manage to win anything.

“I think we should just cave. Actually, it feels better if you just fight a little bit and cave later,” Labrador joked bitterly.

“Sorry,” he deadpanned a moment later. “I’m a little cynical.”

Reid Wilson contributed to this report.

Lori Montgomery covers U.S. economic policy and the federal budget, focusing on efforts to tame the national debt.

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